LMFDB - Embedded newform 2205.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$130.099211563$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} +1.00000 q^{4} +5.00000 q^{5} +21.0000 q^{8} +O(q^{10})$

$q-3.00000 q^{2} +1.00000 q^{4} +5.00000 q^{5} +21.0000 q^{8} -15.0000 q^{10} +45.0000 q^{11} -31.0000 q^{13} -71.0000 q^{16} -96.0000 q^{17} +149.000 q^{19} +5.00000 q^{20} -135.000 q^{22} +141.000 q^{23} +25.0000 q^{25} +93.0000 q^{26} -48.0000 q^{29} -178.000 q^{31} +45.0000 q^{32} +288.000 q^{34} +371.000 q^{37} -447.000 q^{38} +105.000 q^{40} -225.000 q^{41} +344.000 q^{43} +45.0000 q^{44} -423.000 q^{46} -375.000 q^{47} -75.0000 q^{50} -31.0000 q^{52} +663.000 q^{53} +225.000 q^{55} +144.000 q^{58} +60.0000 q^{59} +392.000 q^{61} +534.000 q^{62} +433.000 q^{64} -155.000 q^{65} -280.000 q^{67} -96.0000 q^{68} -258.000 q^{71} +578.000 q^{73} -1113.00 q^{74} +149.000 q^{76} +152.000 q^{79} -355.000 q^{80} +675.000 q^{82} +432.000 q^{83} -480.000 q^{85} -1032.00 q^{86} +945.000 q^{88} +234.000 q^{89} +141.000 q^{92} +1125.00 q^{94} +745.000 q^{95} +1352.00 q^{97} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	21.0000	0.928078
$9$	0	0
$10$	−15.0000	−0.474342
$11$	45.0000	1.23346	0.616728	−	0.787177i	$-0.288458\pi$
$11$	45.0000	1.23346	0.616728	+	0.787177i	$0.288458\pi$
$12$	0	0
$13$	−31.0000	−0.661373	−0.330687	−	0.943741i	$-0.607280\pi$
$13$	−31.0000	−0.661373	−0.330687	+	0.943741i	$0.607280\pi$
$14$	0	0
$15$	0	0
$16$	−71.0000	−1.10938
$17$	−96.0000	−1.36961	−0.684806	−	0.728725i	$-0.740113\pi$
$17$	−96.0000	−1.36961	−0.684806	+	0.728725i	$0.740113\pi$
$18$	0	0
$19$	149.000	1.79910	0.899551	−	0.436815i	$-0.143894\pi$
$19$	149.000	1.79910	0.899551	+	0.436815i	$0.143894\pi$
$20$	5.00000	0.0559017
$21$	0	0
$22$	−135.000	−1.30828
$23$	141.000	1.27828	0.639142	−	0.769089i	$-0.279290\pi$
$23$	141.000	1.27828	0.639142	+	0.769089i	$0.279290\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	93.0000	0.701492
$27$	0	0
$28$	0	0
$29$	−48.0000	−0.307358	−0.153679	−	0.988121i	$-0.549112\pi$
$29$	−48.0000	−0.307358	−0.153679	+	0.988121i	$0.549112\pi$
$30$	0	0
$31$	−178.000	−1.03128	−0.515641	−	0.856805i	$-0.672446\pi$
$31$	−178.000	−1.03128	−0.515641	+	0.856805i	$0.672446\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	288.000	1.45269
$35$	0	0
$36$	0	0
$37$	371.000	1.64843	0.824217	−	0.566275i	$-0.191616\pi$
$37$	371.000	1.64843	0.824217	+	0.566275i	$0.191616\pi$
$38$	−447.000	−1.90824
$39$	0	0
$40$	105.000	0.415049
$41$	−225.000	−0.857051	−0.428526	−	0.903530i	$-0.640967\pi$
$41$	−225.000	−0.857051	−0.428526	+	0.903530i	$0.640967\pi$
$42$	0	0
$43$	344.000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	344.000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	45.0000	0.154182
$45$	0	0
$46$	−423.000	−1.35582
$47$	−375.000	−1.16382	−0.581908	−	0.813254i	$-0.697694\pi$
$47$	−375.000	−1.16382	−0.581908	+	0.813254i	$0.697694\pi$
$48$	0	0
$49$	0	0
$50$	−75.0000	−0.212132
$51$	0	0
$52$	−31.0000	−0.0826717
$53$	663.000	1.71830	0.859151	−	0.511721i	$-0.170992\pi$
$53$	663.000	1.71830	0.859151	+	0.511721i	$0.170992\pi$
$54$	0	0
$55$	225.000	0.551618
$56$	0	0
$57$	0	0
$58$	144.000	0.326002
$59$	60.0000	0.132396	0.0661978	−	0.997807i	$-0.478913\pi$
$59$	60.0000	0.132396	0.0661978	+	0.997807i	$0.478913\pi$
$60$	0	0
$61$	392.000	0.822794	0.411397	−	0.911456i	$-0.365041\pi$
$61$	392.000	0.822794	0.411397	+	0.911456i	$0.365041\pi$
$62$	534.000	1.09384
$63$	0	0
$64$	433.000	0.845703
$65$	−155.000	−0.295775
$66$	0	0
$67$	−280.000	−0.510559	−0.255279	−	0.966867i	$-0.582167\pi$
$67$	−280.000	−0.510559	−0.255279	+	0.966867i	$0.582167\pi$
$68$	−96.0000	−0.171202
$69$	0	0
$70$	0	0
$71$	−258.000	−0.431253	−0.215627	−	0.976476i	$-0.569179\pi$
$71$	−258.000	−0.431253	−0.215627	+	0.976476i	$0.569179\pi$
$72$	0	0
$73$	578.000	0.926709	0.463355	−	0.886173i	$-0.346646\pi$
$73$	578.000	0.926709	0.463355	+	0.886173i	$0.346646\pi$
$74$	−1113.00	−1.74843
$75$	0	0
$76$	149.000	0.224888
$77$	0	0
$78$	0	0
$79$	152.000	0.216473	0.108236	−	0.994125i	$-0.465480\pi$
$79$	152.000	0.216473	0.108236	+	0.994125i	$0.465480\pi$
$80$	−355.000	−0.496128
$81$	0	0
$82$	675.000	0.909040
$83$	432.000	0.571303	0.285652	−	0.958334i	$-0.407790\pi$
$83$	432.000	0.571303	0.285652	+	0.958334i	$0.407790\pi$
$84$	0	0
$85$	−480.000	−0.612510
$86$	−1032.00	−1.29399
$87$	0	0
$88$	945.000	1.14474
$89$	234.000	0.278696	0.139348	−	0.990243i	$-0.455499\pi$
$89$	234.000	0.278696	0.139348	+	0.990243i	$0.455499\pi$
$90$	0	0
$91$	0	0
$92$	141.000	0.159786
$93$	0	0
$94$	1125.00	1.23441
$95$	745.000	0.804583
$96$	0	0
$97$	1352.00	1.41520	0.707602	−	0.706611i	$-0.249777\pi$
$97$	1352.00	1.41520	0.707602	+	0.706611i	$0.249777\pi$
$98$	0	0
$99$	0	0
$100$	25.0000	0.0250000
$101$	−1584.00	−1.56053	−0.780267	−	0.625447i	$-0.784917\pi$
$101$	−1584.00	−1.56053	−0.780267	+	0.625447i	$0.784917\pi$
$102$	0	0
$103$	−976.000	−0.933671	−0.466836	−	0.884344i	$-0.654606\pi$
$103$	−976.000	−0.933671	−0.466836	+	0.884344i	$0.654606\pi$
$104$	−651.000	−0.613806
$105$	0	0
$106$	−1989.00	−1.82254
$107$	−1158.00	−1.04624	−0.523122	−	0.852258i	$-0.675233\pi$
$107$	−1158.00	−1.04624	−0.523122	+	0.852258i	$0.675233\pi$
$108$	0	0
$109$	−970.000	−0.852378	−0.426189	−	0.904634i	$-0.640144\pi$
$109$	−970.000	−0.852378	−0.426189	+	0.904634i	$0.640144\pi$
$110$	−675.000	−0.585079
$111$	0	0
$112$	0	0
$113$	−1758.00	−1.46353	−0.731764	−	0.681558i	$-0.761303\pi$
$113$	−1758.00	−1.46353	−0.731764	+	0.681558i	$0.761303\pi$
$114$	0	0
$115$	705.000	0.571666
$116$	−48.0000	−0.0384197
$117$	0	0
$118$	−180.000	−0.140427
$119$	0	0
$120$	0	0
$121$	694.000	0.521412
$122$	−1176.00	−0.872705
$123$	0	0
$124$	−178.000	−0.128910
$125$	125.000	0.0894427
$126$	0	0
$127$	−457.000	−0.319309	−0.159654	−	0.987173i	$-0.551038\pi$
$127$	−457.000	−0.319309	−0.159654	+	0.987173i	$0.551038\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	465.000	0.313717
$131$	1641.00	1.09446	0.547232	−	0.836981i	$-0.315681\pi$
$131$	1641.00	1.09446	0.547232	+	0.836981i	$0.315681\pi$
$132$	0	0
$133$	0	0
$134$	840.000	0.541529
$135$	0	0
$136$	−2016.00	−1.27111
$137$	1410.00	0.879302	0.439651	−	0.898169i	$-0.355102\pi$
$137$	1410.00	0.879302	0.439651	+	0.898169i	$0.355102\pi$
$138$	0	0
$139$	812.000	0.495489	0.247744	−	0.968825i	$-0.420311\pi$
$139$	812.000	0.495489	0.247744	+	0.968825i	$0.420311\pi$
$140$	0	0
$141$	0	0
$142$	774.000	0.457413
$143$	−1395.00	−0.815775
$144$	0	0
$145$	−240.000	−0.137455
$146$	−1734.00	−0.982924
$147$	0	0
$148$	371.000	0.206054
$149$	−384.000	−0.211131	−0.105565	−	0.994412i	$-0.533665\pi$
$149$	−384.000	−0.211131	−0.105565	+	0.994412i	$0.533665\pi$
$150$	0	0
$151$	2174.00	1.17164	0.585820	−	0.810441i	$-0.300773\pi$
$151$	2174.00	1.17164	0.585820	+	0.810441i	$0.300773\pi$
$152$	3129.00	1.66971
$153$	0	0
$154$	0	0
$155$	−890.000	−0.461203
$156$	0	0
$157$	761.000	0.386843	0.193422	−	0.981116i	$-0.438041\pi$
$157$	761.000	0.386843	0.193422	+	0.981116i	$0.438041\pi$
$158$	−456.000	−0.229604
$159$	0	0
$160$	225.000	0.111174
$161$	0	0
$162$	0	0
$163$	32.0000	0.0153769	0.00768845	−	0.999970i	$-0.497553\pi$
$163$	32.0000	0.0153769	0.00768845	+	0.999970i	$0.497553\pi$
$164$	−225.000	−0.107131
$165$	0	0
$166$	−1296.00	−0.605958
$167$	−81.0000	−0.0375327	−0.0187664	−	0.999824i	$-0.505974\pi$
$167$	−81.0000	−0.0375327	−0.0187664	+	0.999824i	$0.505974\pi$
$168$	0	0
$169$	−1236.00	−0.562585
$170$	1440.00	0.649664
$171$	0	0
$172$	344.000	0.152499
$173$	−741.000	−0.325648	−0.162824	−	0.986655i	$-0.552060\pi$
$173$	−741.000	−0.325648	−0.162824	+	0.986655i	$0.552060\pi$
$174$	0	0
$175$	0	0
$176$	−3195.00	−1.36836
$177$	0	0
$178$	−702.000	−0.295602
$179$	−1857.00	−0.775412	−0.387706	−	0.921783i	$-0.626732\pi$
$179$	−1857.00	−0.775412	−0.387706	+	0.921783i	$0.626732\pi$
$180$	0	0
$181$	−2248.00	−0.923163	−0.461582	−	0.887098i	$-0.652718\pi$
$181$	−2248.00	−0.923163	−0.461582	+	0.887098i	$0.652718\pi$
$182$	0	0
$183$	0	0
$184$	2961.00	1.18635
$185$	1855.00	0.737202
$186$	0	0
$187$	−4320.00	−1.68936
$188$	−375.000	−0.145477
$189$	0	0
$190$	−2235.00	−0.853389
$191$	918.000	0.347770	0.173885	−	0.984766i	$-0.444368\pi$
$191$	918.000	0.347770	0.173885	+	0.984766i	$0.444368\pi$
$192$	0	0
$193$	902.000	0.336411	0.168206	−	0.985752i	$-0.446203\pi$
$193$	902.000	0.336411	0.168206	+	0.985752i	$0.446203\pi$
$194$	−4056.00	−1.50105
$195$	0	0
$196$	0	0
$197$	5469.00	1.97792	0.988960	−	0.148185i	$-0.0473430\pi$
$197$	5469.00	1.97792	0.988960	+	0.148185i	$0.0473430\pi$
$198$	0	0
$199$	−4336.00	−1.54458	−0.772289	−	0.635272i	$-0.780888\pi$
$199$	−4336.00	−1.54458	−0.772289	+	0.635272i	$0.780888\pi$
$200$	525.000	0.185616
$201$	0	0
$202$	4752.00	1.65520
$203$	0	0
$204$	0	0
$205$	−1125.00	−0.383285
$206$	2928.00	0.990308
$207$	0	0
$208$	2201.00	0.733711
$209$	6705.00	2.21911
$210$	0	0
$211$	2045.00	0.667221	0.333610	−	0.942711i	$-0.391733\pi$
$211$	2045.00	0.667221	0.333610	+	0.942711i	$0.391733\pi$
$212$	663.000	0.214788
$213$	0	0
$214$	3474.00	1.10971
$215$	1720.00	0.545595
$216$	0	0
$217$	0	0
$218$	2910.00	0.904083
$219$	0	0
$220$	225.000	0.0689523
$221$	2976.00	0.905825
$222$	0	0
$223$	620.000	0.186181	0.0930903	−	0.995658i	$-0.470325\pi$
$223$	620.000	0.186181	0.0930903	+	0.995658i	$0.470325\pi$
$224$	0	0
$225$	0	0
$226$	5274.00	1.55231
$227$	−5280.00	−1.54381	−0.771907	−	0.635735i	$-0.780697\pi$
$227$	−5280.00	−1.54381	−0.771907	+	0.635735i	$0.780697\pi$
$228$	0	0
$229$	3302.00	0.952848	0.476424	−	0.879216i	$-0.341933\pi$
$229$	3302.00	0.952848	0.476424	+	0.879216i	$0.341933\pi$
$230$	−2115.00	−0.606343
$231$	0	0
$232$	−1008.00	−0.285252
$233$	−1818.00	−0.511164	−0.255582	−	0.966787i	$-0.582267\pi$
$233$	−1818.00	−0.511164	−0.255582	+	0.966787i	$0.582267\pi$
$234$	0	0
$235$	−1875.00	−0.520475
$236$	60.0000	0.0165494
$237$	0	0
$238$	0	0
$239$	−1638.00	−0.443320	−0.221660	−	0.975124i	$-0.571147\pi$
$239$	−1638.00	−0.443320	−0.221660	+	0.975124i	$0.571147\pi$
$240$	0	0
$241$	1391.00	0.371793	0.185897	−	0.982569i	$-0.440481\pi$
$241$	1391.00	0.371793	0.185897	+	0.982569i	$0.440481\pi$
$242$	−2082.00	−0.553041
$243$	0	0
$244$	392.000	0.102849
$245$	0	0
$246$	0	0
$247$	−4619.00	−1.18988
$248$	−3738.00	−0.957110
$249$	0	0
$250$	−375.000	−0.0948683
$251$	2655.00	0.667658	0.333829	−	0.942634i	$-0.391659\pi$
$251$	2655.00	0.667658	0.333829	+	0.942634i	$0.391659\pi$
$252$	0	0
$253$	6345.00	1.57671
$254$	1371.00	0.338678
$255$	0	0
$256$	1513.00	0.369385
$257$	2220.00	0.538832	0.269416	−	0.963024i	$-0.413169\pi$
$257$	2220.00	0.538832	0.269416	+	0.963024i	$0.413169\pi$
$258$	0	0
$259$	0	0
$260$	−155.000	−0.0369719
$261$	0	0
$262$	−4923.00	−1.16085
$263$	−912.000	−0.213826	−0.106913	−	0.994268i	$-0.534097\pi$
$263$	−912.000	−0.213826	−0.106913	+	0.994268i	$0.534097\pi$
$264$	0	0
$265$	3315.00	0.768448
$266$	0	0
$267$	0	0
$268$	−280.000	−0.0638199
$269$	−156.000	−0.0353587	−0.0176793	−	0.999844i	$-0.505628\pi$
$269$	−156.000	−0.0353587	−0.0176793	+	0.999844i	$0.505628\pi$
$270$	0	0
$271$	−4192.00	−0.939653	−0.469826	−	0.882759i	$-0.655683\pi$
$271$	−4192.00	−0.939653	−0.469826	+	0.882759i	$0.655683\pi$
$272$	6816.00	1.51941
$273$	0	0
$274$	−4230.00	−0.932641
$275$	1125.00	0.246691
$276$	0	0
$277$	−5950.00	−1.29062	−0.645308	−	0.763922i	$-0.723271\pi$
$277$	−5950.00	−1.29062	−0.645308	+	0.763922i	$0.723271\pi$
$278$	−2436.00	−0.525545
$279$	0	0
$280$	0	0
$281$	−4251.00	−0.902468	−0.451234	−	0.892406i	$-0.649016\pi$
$281$	−4251.00	−0.902468	−0.451234	+	0.892406i	$0.649016\pi$
$282$	0	0
$283$	−4438.00	−0.932197	−0.466098	−	0.884733i	$-0.654341\pi$
$283$	−4438.00	−0.932197	−0.466098	+	0.884733i	$0.654341\pi$
$284$	−258.000	−0.0539066
$285$	0	0
$286$	4185.00	0.865260
$287$	0	0
$288$	0	0
$289$	4303.00	0.875840
$290$	720.000	0.145793
$291$	0	0
$292$	578.000	0.115839
$293$	1017.00	0.202777	0.101389	−	0.994847i	$-0.467671\pi$
$293$	1017.00	0.202777	0.101389	+	0.994847i	$0.467671\pi$
$294$	0	0
$295$	300.000	0.0592091
$296$	7791.00	1.52987
$297$	0	0
$298$	1152.00	0.223938
$299$	−4371.00	−0.845423
$300$	0	0
$301$	0	0
$302$	−6522.00	−1.24271
$303$	0	0
$304$	−10579.0	−1.99588
$305$	1960.00	0.367965
$306$	0	0
$307$	5204.00	0.967453	0.483726	−	0.875219i	$-0.339283\pi$
$307$	5204.00	0.967453	0.483726	+	0.875219i	$0.339283\pi$
$308$	0	0
$309$	0	0
$310$	2670.00	0.489180
$311$	60.0000	0.0109398	0.00546992	−	0.999985i	$-0.498259\pi$
$311$	60.0000	0.0109398	0.00546992	+	0.999985i	$0.498259\pi$
$312$	0	0
$313$	7148.00	1.29083	0.645413	−	0.763833i	$-0.276685\pi$
$313$	7148.00	1.29083	0.645413	+	0.763833i	$0.276685\pi$
$314$	−2283.00	−0.410309
$315$	0	0
$316$	152.000	0.0270591
$317$	7338.00	1.30014	0.650068	−	0.759876i	$-0.274740\pi$
$317$	7338.00	1.30014	0.650068	+	0.759876i	$0.274740\pi$
$318$	0	0
$319$	−2160.00	−0.379112
$320$	2165.00	0.378210
$321$	0	0
$322$	0	0
$323$	−14304.0	−2.46407
$324$	0	0
$325$	−775.000	−0.132275
$326$	−96.0000	−0.0163097
$327$	0	0
$328$	−4725.00	−0.795410
$329$	0	0
$330$	0	0
$331$	2411.00	0.400364	0.200182	−	0.979759i	$-0.435847\pi$
$331$	2411.00	0.400364	0.200182	+	0.979759i	$0.435847\pi$
$332$	432.000	0.0714129
$333$	0	0
$334$	243.000	0.0398095
$335$	−1400.00	−0.228329
$336$	0	0
$337$	9524.00	1.53948	0.769741	−	0.638356i	$-0.220385\pi$
$337$	9524.00	1.53948	0.769741	+	0.638356i	$0.220385\pi$
$338$	3708.00	0.596712
$339$	0	0
$340$	−480.000	−0.0765637
$341$	−8010.00	−1.27204
$342$	0	0
$343$	0	0
$344$	7224.00	1.13224
$345$	0	0
$346$	2223.00	0.345402
$347$	4290.00	0.663687	0.331843	−	0.943335i	$-0.392329\pi$
$347$	4290.00	0.663687	0.331843	+	0.943335i	$0.392329\pi$
$348$	0	0
$349$	1892.00	0.290190	0.145095	−	0.989418i	$-0.453651\pi$
$349$	1892.00	0.290190	0.145095	+	0.989418i	$0.453651\pi$
$350$	0	0
$351$	0	0
$352$	2025.00	0.306627
$353$	72.0000	0.0108560	0.00542801	−	0.999985i	$-0.498272\pi$
$353$	72.0000	0.0108560	0.00542801	+	0.999985i	$0.498272\pi$
$354$	0	0
$355$	−1290.00	−0.192862
$356$	234.000	0.0348370
$357$	0	0
$358$	5571.00	0.822448
$359$	6180.00	0.908546	0.454273	−	0.890863i	$-0.349899\pi$
$359$	6180.00	0.908546	0.454273	+	0.890863i	$0.349899\pi$
$360$	0	0
$361$	15342.0	2.23677
$362$	6744.00	0.979162
$363$	0	0
$364$	0	0
$365$	2890.00	0.414437
$366$	0	0
$367$	9281.00	1.32007	0.660033	−	0.751237i	$-0.270542\pi$
$367$	9281.00	1.32007	0.660033	+	0.751237i	$0.270542\pi$
$368$	−10011.0	−1.41810
$369$	0	0
$370$	−5565.00	−0.781920
$371$	0	0
$372$	0	0
$373$	10382.0	1.44118	0.720589	−	0.693362i	$-0.243871\pi$
$373$	10382.0	1.44118	0.720589	+	0.693362i	$0.243871\pi$
$374$	12960.0	1.79183
$375$	0	0
$376$	−7875.00	−1.08011
$377$	1488.00	0.203278
$378$	0	0
$379$	9965.00	1.35057	0.675287	−	0.737555i	$-0.264020\pi$
$379$	9965.00	1.35057	0.675287	+	0.737555i	$0.264020\pi$
$380$	745.000	0.100573
$381$	0	0
$382$	−2754.00	−0.368866
$383$	3465.00	0.462280	0.231140	−	0.972921i	$-0.425754\pi$
$383$	3465.00	0.462280	0.231140	+	0.972921i	$0.425754\pi$
$384$	0	0
$385$	0	0
$386$	−2706.00	−0.356818
$387$	0	0
$388$	1352.00	0.176901
$389$	1614.00	0.210368	0.105184	−	0.994453i	$-0.466457\pi$
$389$	1614.00	0.210368	0.105184	+	0.994453i	$0.466457\pi$
$390$	0	0
$391$	−13536.0	−1.75075
$392$	0	0
$393$	0	0
$394$	−16407.0	−2.09790
$395$	760.000	0.0968095
$396$	0	0
$397$	3062.00	0.387097	0.193548	−	0.981091i	$-0.438000\pi$
$397$	3062.00	0.387097	0.193548	+	0.981091i	$0.438000\pi$
$398$	13008.0	1.63827
$399$	0	0
$400$	−1775.00	−0.221875
$401$	−543.000	−0.0676213	−0.0338106	−	0.999428i	$-0.510764\pi$
$401$	−543.000	−0.0676213	−0.0338106	+	0.999428i	$0.510764\pi$
$402$	0	0
$403$	5518.00	0.682062
$404$	−1584.00	−0.195067
$405$	0	0
$406$	0	0
$407$	16695.0	2.03327
$408$	0	0
$409$	8210.00	0.992563	0.496282	−	0.868162i	$-0.334698\pi$
$409$	8210.00	0.992563	0.496282	+	0.868162i	$0.334698\pi$
$410$	3375.00	0.406535
$411$	0	0
$412$	−976.000	−0.116709
$413$	0	0
$414$	0	0
$415$	2160.00	0.255495
$416$	−1395.00	−0.164412
$417$	0	0
$418$	−20115.0	−2.35372
$419$	−3471.00	−0.404700	−0.202350	−	0.979313i	$-0.564858\pi$
$419$	−3471.00	−0.404700	−0.202350	+	0.979313i	$0.564858\pi$
$420$	0	0
$421$	−6250.00	−0.723531	−0.361765	−	0.932269i	$-0.617826\pi$
$421$	−6250.00	−0.723531	−0.361765	+	0.932269i	$0.617826\pi$
$422$	−6135.00	−0.707695
$423$	0	0
$424$	13923.0	1.59472
$425$	−2400.00	−0.273923
$426$	0	0
$427$	0	0
$428$	−1158.00	−0.130780
$429$	0	0
$430$	−5160.00	−0.578691
$431$	9192.00	1.02729	0.513646	−	0.858002i	$-0.328294\pi$
$431$	9192.00	1.02729	0.513646	+	0.858002i	$0.328294\pi$
$432$	0	0
$433$	11216.0	1.24482	0.622409	−	0.782692i	$-0.286154\pi$
$433$	11216.0	1.24482	0.622409	+	0.782692i	$0.286154\pi$
$434$	0	0
$435$	0	0
$436$	−970.000	−0.106547
$437$	21009.0	2.29976
$438$	0	0
$439$	17336.0	1.88474	0.942371	−	0.334568i	$-0.108591\pi$
$439$	17336.0	1.88474	0.942371	+	0.334568i	$0.108591\pi$
$440$	4725.00	0.511944
$441$	0	0
$442$	−8928.00	−0.960773
$443$	−6312.00	−0.676957	−0.338479	−	0.940974i	$-0.609912\pi$
$443$	−6312.00	−0.676957	−0.338479	+	0.940974i	$0.609912\pi$
$444$	0	0
$445$	1170.00	0.124637
$446$	−1860.00	−0.197474
$447$	0	0
$448$	0	0
$449$	10281.0	1.08060	0.540301	−	0.841472i	$-0.318310\pi$
$449$	10281.0	1.08060	0.540301	+	0.841472i	$0.318310\pi$
$450$	0	0
$451$	−10125.0	−1.05713
$452$	−1758.00	−0.182941
$453$	0	0
$454$	15840.0	1.63746
$455$	0	0
$456$	0	0
$457$	13046.0	1.33537	0.667687	−	0.744442i	$-0.267284\pi$
$457$	13046.0	1.33537	0.667687	+	0.744442i	$0.267284\pi$
$458$	−9906.00	−1.01065
$459$	0	0
$460$	705.000	0.0714582
$461$	−16428.0	−1.65971	−0.829857	−	0.557976i	$-0.811578\pi$
$461$	−16428.0	−1.65971	−0.829857	+	0.557976i	$0.811578\pi$
$462$	0	0
$463$	7451.00	0.747899	0.373950	−	0.927449i	$-0.378003\pi$
$463$	7451.00	0.747899	0.373950	+	0.927449i	$0.378003\pi$
$464$	3408.00	0.340975
$465$	0	0
$466$	5454.00	0.542171
$467$	−9984.00	−0.989303	−0.494651	−	0.869091i	$-0.664704\pi$
$467$	−9984.00	−0.989303	−0.494651	+	0.869091i	$0.664704\pi$
$468$	0	0
$469$	0	0
$470$	5625.00	0.552047
$471$	0	0
$472$	1260.00	0.122873
$473$	15480.0	1.50480
$474$	0	0
$475$	3725.00	0.359820
$476$	0	0
$477$	0	0
$478$	4914.00	0.470212
$479$	−3078.00	−0.293606	−0.146803	−	0.989166i	$-0.546898\pi$
$479$	−3078.00	−0.293606	−0.146803	+	0.989166i	$0.546898\pi$
$480$	0	0
$481$	−11501.0	−1.09023
$482$	−4173.00	−0.394346
$483$	0	0
$484$	694.000	0.0651766
$485$	6760.00	0.632899
$486$	0	0
$487$	5744.00	0.534467	0.267234	−	0.963632i	$-0.413890\pi$
$487$	5744.00	0.534467	0.267234	+	0.963632i	$0.413890\pi$
$488$	8232.00	0.763617
$489$	0	0
$490$	0	0
$491$	−2004.00	−0.184194	−0.0920970	−	0.995750i	$-0.529357\pi$
$491$	−2004.00	−0.184194	−0.0920970	+	0.995750i	$0.529357\pi$
$492$	0	0
$493$	4608.00	0.420961
$494$	13857.0	1.26206
$495$	0	0
$496$	12638.0	1.14408
$497$	0	0
$498$	0	0
$499$	−12136.0	−1.08874	−0.544371	−	0.838845i	$-0.683232\pi$
$499$	−12136.0	−1.08874	−0.544371	+	0.838845i	$0.683232\pi$
$500$	125.000	0.0111803
$501$	0	0
$502$	−7965.00	−0.708158
$503$	660.000	0.0585049	0.0292524	−	0.999572i	$-0.490687\pi$
$503$	660.000	0.0585049	0.0292524	+	0.999572i	$0.490687\pi$
$504$	0	0
$505$	−7920.00	−0.697892
$506$	−19035.0	−1.67235
$507$	0	0
$508$	−457.000	−0.0399136
$509$	10242.0	0.891883	0.445942	−	0.895062i	$-0.352869\pi$
$509$	10242.0	0.891883	0.445942	+	0.895062i	$0.352869\pi$
$510$	0	0
$511$	0	0
$512$	8733.00	0.753804
$513$	0	0
$514$	−6660.00	−0.571518
$515$	−4880.00	−0.417550
$516$	0	0
$517$	−16875.0	−1.43552
$518$	0	0
$519$	0	0
$520$	−3255.00	−0.274502
$521$	−6939.00	−0.583499	−0.291750	−	0.956495i	$-0.594237\pi$
$521$	−6939.00	−0.583499	−0.291750	+	0.956495i	$0.594237\pi$
$522$	0	0
$523$	−4822.00	−0.403157	−0.201579	−	0.979472i	$-0.564607\pi$
$523$	−4822.00	−0.403157	−0.201579	+	0.979472i	$0.564607\pi$
$524$	1641.00	0.136808
$525$	0	0
$526$	2736.00	0.226797
$527$	17088.0	1.41246
$528$	0	0
$529$	7714.00	0.634010
$530$	−9945.00	−0.815063
$531$	0	0
$532$	0	0
$533$	6975.00	0.566831
$534$	0	0
$535$	−5790.00	−0.467894
$536$	−5880.00	−0.473838
$537$	0	0
$538$	468.000	0.0375036
$539$	0	0
$540$	0	0
$541$	−1378.00	−0.109510	−0.0547549	−	0.998500i	$-0.517438\pi$
$541$	−1378.00	−0.109510	−0.0547549	+	0.998500i	$0.517438\pi$
$542$	12576.0	0.996652
$543$	0	0
$544$	−4320.00	−0.340475
$545$	−4850.00	−0.381195
$546$	0	0
$547$	20504.0	1.60272	0.801360	−	0.598183i	$-0.204110\pi$
$547$	20504.0	1.60272	0.801360	+	0.598183i	$0.204110\pi$
$548$	1410.00	0.109913
$549$	0	0
$550$	−3375.00	−0.261655
$551$	−7152.00	−0.552968
$552$	0	0
$553$	0	0
$554$	17850.0	1.36891
$555$	0	0
$556$	812.000	0.0619361
$557$	−3243.00	−0.246697	−0.123349	−	0.992363i	$-0.539363\pi$
$557$	−3243.00	−0.246697	−0.123349	+	0.992363i	$0.539363\pi$
$558$	0	0
$559$	−10664.0	−0.806868
$560$	0	0
$561$	0	0
$562$	12753.0	0.957211
$563$	7734.00	0.578951	0.289475	−	0.957185i	$-0.406519\pi$
$563$	7734.00	0.578951	0.289475	+	0.957185i	$0.406519\pi$
$564$	0	0
$565$	−8790.00	−0.654510
$566$	13314.0	0.988744
$567$	0	0
$568$	−5418.00	−0.400236
$569$	17037.0	1.25523	0.627617	−	0.778522i	$-0.284030\pi$
$569$	17037.0	1.25523	0.627617	+	0.778522i	$0.284030\pi$
$570$	0	0
$571$	−22888.0	−1.67747	−0.838733	−	0.544543i	$-0.816703\pi$
$571$	−22888.0	−1.67747	−0.838733	+	0.544543i	$0.816703\pi$
$572$	−1395.00	−0.101972
$573$	0	0
$574$	0	0
$575$	3525.00	0.255657
$576$	0	0
$577$	20954.0	1.51183	0.755915	−	0.654669i	$-0.227192\pi$
$577$	20954.0	1.51183	0.755915	+	0.654669i	$0.227192\pi$
$578$	−12909.0	−0.928968
$579$	0	0
$580$	−240.000	−0.0171818
$581$	0	0
$582$	0	0
$583$	29835.0	2.11945
$584$	12138.0	0.860058
$585$	0	0
$586$	−3051.00	−0.215078
$587$	−2742.00	−0.192801	−0.0964007	−	0.995343i	$-0.530733\pi$
$587$	−2742.00	−0.192801	−0.0964007	+	0.995343i	$0.530733\pi$
$588$	0	0
$589$	−26522.0	−1.85538
$590$	−900.000	−0.0628007
$591$	0	0
$592$	−26341.0	−1.82873
$593$	7176.00	0.496936	0.248468	−	0.968640i	$-0.420073\pi$
$593$	7176.00	0.496936	0.248468	+	0.968640i	$0.420073\pi$
$594$	0	0
$595$	0	0
$596$	−384.000	−0.0263914
$597$	0	0
$598$	13113.0	0.896706
$599$	−12234.0	−0.834504	−0.417252	−	0.908791i	$-0.637007\pi$
$599$	−12234.0	−0.834504	−0.417252	+	0.908791i	$0.637007\pi$
$600$	0	0
$601$	4790.00	0.325105	0.162553	−	0.986700i	$-0.448027\pi$
$601$	4790.00	0.325105	0.162553	+	0.986700i	$0.448027\pi$
$602$	0	0
$603$	0	0
$604$	2174.00	0.146455
$605$	3470.00	0.233183
$606$	0	0
$607$	26411.0	1.76605	0.883023	−	0.469330i	$-0.155505\pi$
$607$	26411.0	1.76605	0.883023	+	0.469330i	$0.155505\pi$
$608$	6705.00	0.447243
$609$	0	0
$610$	−5880.00	−0.390286
$611$	11625.0	0.769717
$612$	0	0
$613$	22007.0	1.45001	0.725004	−	0.688745i	$-0.241838\pi$
$613$	22007.0	1.45001	0.725004	+	0.688745i	$0.241838\pi$
$614$	−15612.0	−1.02614
$615$	0	0
$616$	0	0
$617$	−24588.0	−1.60434	−0.802168	−	0.597098i	$-0.796320\pi$
$617$	−24588.0	−1.60434	−0.802168	+	0.597098i	$0.796320\pi$
$618$	0	0
$619$	−22003.0	−1.42872	−0.714358	−	0.699780i	$-0.753281\pi$
$619$	−22003.0	−1.42872	−0.714358	+	0.699780i	$0.753281\pi$
$620$	−890.000	−0.0576504
$621$	0	0
$622$	−180.000	−0.0116034
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−21444.0	−1.36913
$627$	0	0
$628$	761.000	0.0483554
$629$	−35616.0	−2.25772
$630$	0	0
$631$	2390.00	0.150784	0.0753918	−	0.997154i	$-0.475979\pi$
$631$	2390.00	0.150784	0.0753918	+	0.997154i	$0.475979\pi$
$632$	3192.00	0.200903
$633$	0	0
$634$	−22014.0	−1.37900
$635$	−2285.00	−0.142799
$636$	0	0
$637$	0	0
$638$	6480.00	0.402109
$639$	0	0
$640$	−8295.00	−0.512326
$641$	−10569.0	−0.651249	−0.325624	−	0.945499i	$-0.605574\pi$
$641$	−10569.0	−0.651249	−0.325624	+	0.945499i	$0.605574\pi$
$642$	0	0
$643$	−6418.00	−0.393626	−0.196813	−	0.980441i	$-0.563059\pi$
$643$	−6418.00	−0.393626	−0.196813	+	0.980441i	$0.563059\pi$
$644$	0	0
$645$	0	0
$646$	42912.0	2.61354
$647$	−26811.0	−1.62913	−0.814567	−	0.580070i	$-0.803025\pi$
$647$	−26811.0	−1.62913	−0.814567	+	0.580070i	$0.803025\pi$
$648$	0	0
$649$	2700.00	0.163304
$650$	2325.00	0.140298
$651$	0	0
$652$	32.0000	0.00192211
$653$	−483.000	−0.0289453	−0.0144726	−	0.999895i	$-0.504607\pi$
$653$	−483.000	−0.0289453	−0.0144726	+	0.999895i	$0.504607\pi$
$654$	0	0
$655$	8205.00	0.489459
$656$	15975.0	0.950791
$657$	0	0
$658$	0	0
$659$	28236.0	1.66907	0.834536	−	0.550953i	$-0.185736\pi$
$659$	28236.0	1.66907	0.834536	+	0.550953i	$0.185736\pi$
$660$	0	0
$661$	−664.000	−0.0390720	−0.0195360	−	0.999809i	$-0.506219\pi$
$661$	−664.000	−0.0390720	−0.0195360	+	0.999809i	$0.506219\pi$
$662$	−7233.00	−0.424650
$663$	0	0
$664$	9072.00	0.530214
$665$	0	0
$666$	0	0
$667$	−6768.00	−0.392891
$668$	−81.0000	−0.00469159
$669$	0	0
$670$	4200.00	0.242179
$671$	17640.0	1.01488
$672$	0	0
$673$	13508.0	0.773693	0.386846	−	0.922144i	$-0.373564\pi$
$673$	13508.0	0.773693	0.386846	+	0.922144i	$0.373564\pi$
$674$	−28572.0	−1.63287
$675$	0	0
$676$	−1236.00	−0.0703232
$677$	−8103.00	−0.460005	−0.230003	−	0.973190i	$-0.573873\pi$
$677$	−8103.00	−0.460005	−0.230003	+	0.973190i	$0.573873\pi$
$678$	0	0
$679$	0	0
$680$	−10080.0	−0.568456
$681$	0	0
$682$	24030.0	1.34920
$683$	31020.0	1.73784	0.868922	−	0.494949i	$-0.164813\pi$
$683$	31020.0	1.73784	0.868922	+	0.494949i	$0.164813\pi$
$684$	0	0
$685$	7050.00	0.393236
$686$	0	0
$687$	0	0
$688$	−24424.0	−1.35342
$689$	−20553.0	−1.13644
$690$	0	0
$691$	6620.00	0.364452	0.182226	−	0.983257i	$-0.441670\pi$
$691$	6620.00	0.364452	0.182226	+	0.983257i	$0.441670\pi$
$692$	−741.000	−0.0407061
$693$	0	0
$694$	−12870.0	−0.703946
$695$	4060.00	0.221589
$696$	0	0
$697$	21600.0	1.17383
$698$	−5676.00	−0.307793
$699$	0	0
$700$	0	0
$701$	9186.00	0.494936	0.247468	−	0.968896i	$-0.420401\pi$
$701$	9186.00	0.494936	0.247468	+	0.968896i	$0.420401\pi$
$702$	0	0
$703$	55279.0	2.96570
$704$	19485.0	1.04314
$705$	0	0
$706$	−216.000	−0.0115145
$707$	0	0
$708$	0	0
$709$	9524.00	0.504487	0.252244	−	0.967664i	$-0.418832\pi$
$709$	9524.00	0.504487	0.252244	+	0.967664i	$0.418832\pi$
$710$	3870.00	0.204561
$711$	0	0
$712$	4914.00	0.258652
$713$	−25098.0	−1.31827
$714$	0	0
$715$	−6975.00	−0.364825
$716$	−1857.00	−0.0969265
$717$	0	0
$718$	−18540.0	−0.963658
$719$	4086.00	0.211936	0.105968	−	0.994370i	$-0.466206\pi$
$719$	4086.00	0.211936	0.105968	+	0.994370i	$0.466206\pi$
$720$	0	0
$721$	0	0
$722$	−46026.0	−2.37245
$723$	0	0
$724$	−2248.00	−0.115395
$725$	−1200.00	−0.0614716
$726$	0	0
$727$	32459.0	1.65590	0.827949	−	0.560804i	$-0.189508\pi$
$727$	32459.0	1.65590	0.827949	+	0.560804i	$0.189508\pi$
$728$	0	0
$729$	0	0
$730$	−8670.00	−0.439577
$731$	−33024.0	−1.67091
$732$	0	0
$733$	8363.00	0.421411	0.210706	−	0.977550i	$-0.432424\pi$
$733$	8363.00	0.421411	0.210706	+	0.977550i	$0.432424\pi$
$734$	−27843.0	−1.40014
$735$	0	0
$736$	6345.00	0.317771
$737$	−12600.0	−0.629752
$738$	0	0
$739$	−34171.0	−1.70095	−0.850474	−	0.526017i	$-0.823685\pi$
$739$	−34171.0	−1.70095	−0.850474	+	0.526017i	$0.823685\pi$
$740$	1855.00	0.0921502
$741$	0	0
$742$	0	0
$743$	15723.0	0.776340	0.388170	−	0.921588i	$-0.373107\pi$
$743$	15723.0	0.776340	0.388170	+	0.921588i	$0.373107\pi$
$744$	0	0
$745$	−1920.00	−0.0944206
$746$	−31146.0	−1.52860
$747$	0	0
$748$	−4320.00	−0.211170
$749$	0	0
$750$	0	0
$751$	−7222.00	−0.350911	−0.175456	−	0.984487i	$-0.556140\pi$
$751$	−7222.00	−0.350911	−0.175456	+	0.984487i	$0.556140\pi$
$752$	26625.0	1.29111
$753$	0	0
$754$	−4464.00	−0.215609
$755$	10870.0	0.523973
$756$	0	0
$757$	36266.0	1.74123	0.870614	−	0.491966i	$-0.163722\pi$
$757$	36266.0	1.74123	0.870614	+	0.491966i	$0.163722\pi$
$758$	−29895.0	−1.43250
$759$	0	0
$760$	15645.0	0.746716
$761$	−31119.0	−1.48234	−0.741171	−	0.671316i	$-0.765729\pi$
$761$	−31119.0	−1.48234	−0.741171	+	0.671316i	$0.765729\pi$
$762$	0	0
$763$	0	0
$764$	918.000	0.0434713
$765$	0	0
$766$	−10395.0	−0.490322
$767$	−1860.00	−0.0875629
$768$	0	0
$769$	875.000	0.0410316	0.0205158	−	0.999790i	$-0.493469\pi$
$769$	875.000	0.0410316	0.0205158	+	0.999790i	$0.493469\pi$
$770$	0	0
$771$	0	0
$772$	902.000	0.0420514
$773$	−7821.00	−0.363909	−0.181955	−	0.983307i	$-0.558242\pi$
$773$	−7821.00	−0.363909	−0.181955	+	0.983307i	$0.558242\pi$
$774$	0	0
$775$	−4450.00	−0.206256
$776$	28392.0	1.31342
$777$	0	0
$778$	−4842.00	−0.223129
$779$	−33525.0	−1.54192
$780$	0	0
$781$	−11610.0	−0.531931
$782$	40608.0	1.85696
$783$	0	0
$784$	0	0
$785$	3805.00	0.173002
$786$	0	0
$787$	−3586.00	−0.162423	−0.0812116	−	0.996697i	$-0.525879\pi$
$787$	−3586.00	−0.162423	−0.0812116	+	0.996697i	$0.525879\pi$
$788$	5469.00	0.247240
$789$	0	0
$790$	−2280.00	−0.102682
$791$	0	0
$792$	0	0
$793$	−12152.0	−0.544174
$794$	−9186.00	−0.410578
$795$	0	0
$796$	−4336.00	−0.193072
$797$	−14790.0	−0.657326	−0.328663	−	0.944447i	$-0.606598\pi$
$797$	−14790.0	−0.657326	−0.328663	+	0.944447i	$0.606598\pi$
$798$	0	0
$799$	36000.0	1.59398
$800$	1125.00	0.0497184
$801$	0	0
$802$	1629.00	0.0717232
$803$	26010.0	1.14305
$804$	0	0
$805$	0	0
$806$	−16554.0	−0.723436
$807$	0	0
$808$	−33264.0	−1.44830
$809$	35259.0	1.53231	0.766156	−	0.642655i	$-0.222167\pi$
$809$	35259.0	1.53231	0.766156	+	0.642655i	$0.222167\pi$
$810$	0	0
$811$	−6607.00	−0.286070	−0.143035	−	0.989718i	$-0.545686\pi$
$811$	−6607.00	−0.286070	−0.143035	+	0.989718i	$0.545686\pi$
$812$	0	0
$813$	0	0
$814$	−50085.0	−2.15661
$815$	160.000	0.00687676
$816$	0	0
$817$	51256.0	2.19488
$818$	−24630.0	−1.05277
$819$	0	0
$820$	−1125.00	−0.0479106
$821$	18918.0	0.804193	0.402097	−	0.915597i	$-0.368282\pi$
$821$	18918.0	0.804193	0.402097	+	0.915597i	$0.368282\pi$
$822$	0	0
$823$	−40120.0	−1.69927	−0.849633	−	0.527375i	$-0.823176\pi$
$823$	−40120.0	−1.69927	−0.849633	+	0.527375i	$0.823176\pi$
$824$	−20496.0	−0.866519
$825$	0	0
$826$	0	0
$827$	282.000	0.0118574	0.00592872	−	0.999982i	$-0.498113\pi$
$827$	282.000	0.0118574	0.00592872	+	0.999982i	$0.498113\pi$
$828$	0	0
$829$	1880.00	0.0787637	0.0393818	−	0.999224i	$-0.487461\pi$
$829$	1880.00	0.0787637	0.0393818	+	0.999224i	$0.487461\pi$
$830$	−6480.00	−0.270993
$831$	0	0
$832$	−13423.0	−0.559325
$833$	0	0
$834$	0	0
$835$	−405.000	−0.0167852
$836$	6705.00	0.277389
$837$	0	0
$838$	10413.0	0.429250
$839$	10728.0	0.441444	0.220722	−	0.975337i	$-0.429159\pi$
$839$	10728.0	0.441444	0.220722	+	0.975337i	$0.429159\pi$
$840$	0	0
$841$	−22085.0	−0.905531
$842$	18750.0	0.767420
$843$	0	0
$844$	2045.00	0.0834026
$845$	−6180.00	−0.251596
$846$	0	0
$847$	0	0
$848$	−47073.0	−1.90624
$849$	0	0
$850$	7200.00	0.290539
$851$	52311.0	2.10717
$852$	0	0
$853$	−40777.0	−1.63679	−0.818393	−	0.574659i	$-0.805135\pi$
$853$	−40777.0	−1.63679	−0.818393	+	0.574659i	$0.805135\pi$
$854$	0	0
$855$	0	0
$856$	−24318.0	−0.970995
$857$	45366.0	1.80825	0.904127	−	0.427265i	$-0.140523\pi$
$857$	45366.0	1.80825	0.904127	+	0.427265i	$0.140523\pi$
$858$	0	0
$859$	15572.0	0.618521	0.309261	−	0.950977i	$-0.399918\pi$
$859$	15572.0	0.618521	0.309261	+	0.950977i	$0.399918\pi$
$860$	1720.00	0.0681994
$861$	0	0
$862$	−27576.0	−1.08961
$863$	−42807.0	−1.68849	−0.844245	−	0.535957i	$-0.819951\pi$
$863$	−42807.0	−1.68849	−0.844245	+	0.535957i	$0.819951\pi$
$864$	0	0
$865$	−3705.00	−0.145634
$866$	−33648.0	−1.32033
$867$	0	0
$868$	0	0
$869$	6840.00	0.267009
$870$	0	0
$871$	8680.00	0.337670
$872$	−20370.0	−0.791073
$873$	0	0
$874$	−63027.0	−2.43927
$875$	0	0
$876$	0	0
$877$	−50311.0	−1.93715	−0.968576	−	0.248719i	$-0.919990\pi$
$877$	−50311.0	−1.93715	−0.968576	+	0.248719i	$0.919990\pi$
$878$	−52008.0	−1.99907
$879$	0	0
$880$	−15975.0	−0.611951
$881$	−29967.0	−1.14599	−0.572993	−	0.819560i	$-0.694218\pi$
$881$	−29967.0	−1.14599	−0.572993	+	0.819560i	$0.694218\pi$
$882$	0	0
$883$	−4018.00	−0.153133	−0.0765665	−	0.997064i	$-0.524396\pi$
$883$	−4018.00	−0.153133	−0.0765665	+	0.997064i	$0.524396\pi$
$884$	2976.00	0.113228
$885$	0	0
$886$	18936.0	0.718022
$887$	21036.0	0.796302	0.398151	−	0.917320i	$-0.369652\pi$
$887$	21036.0	0.796302	0.398151	+	0.917320i	$0.369652\pi$
$888$	0	0
$889$	0	0
$890$	−3510.00	−0.132197
$891$	0	0
$892$	620.000	0.0232726
$893$	−55875.0	−2.09382
$894$	0	0
$895$	−9285.00	−0.346775
$896$	0	0
$897$	0	0
$898$	−30843.0	−1.14615
$899$	8544.00	0.316973
$900$	0	0
$901$	−63648.0	−2.35341
$902$	30375.0	1.12126
$903$	0	0
$904$	−36918.0	−1.35827
$905$	−11240.0	−0.412851
$906$	0	0
$907$	30518.0	1.11724	0.558618	−	0.829425i	$-0.311332\pi$
$907$	30518.0	1.11724	0.558618	+	0.829425i	$0.311332\pi$
$908$	−5280.00	−0.192977
$909$	0	0
$910$	0	0
$911$	10866.0	0.395177	0.197589	−	0.980285i	$-0.436689\pi$
$911$	10866.0	0.395177	0.197589	+	0.980285i	$0.436689\pi$
$912$	0	0
$913$	19440.0	0.704677
$914$	−39138.0	−1.41638
$915$	0	0
$916$	3302.00	0.119106
$917$	0	0
$918$	0	0
$919$	−5884.00	−0.211203	−0.105601	−	0.994409i	$-0.533677\pi$
$919$	−5884.00	−0.211203	−0.105601	+	0.994409i	$0.533677\pi$
$920$	14805.0	0.530550
$921$	0	0
$922$	49284.0	1.76039
$923$	7998.00	0.285219
$924$	0	0
$925$	9275.00	0.329687
$926$	−22353.0	−0.793267
$927$	0	0
$928$	−2160.00	−0.0764068
$929$	21441.0	0.757219	0.378609	−	0.925557i	$-0.376402\pi$
$929$	21441.0	0.757219	0.378609	+	0.925557i	$0.376402\pi$
$930$	0	0
$931$	0	0
$932$	−1818.00	−0.0638955
$933$	0	0
$934$	29952.0	1.04931
$935$	−21600.0	−0.755503
$936$	0	0
$937$	9596.00	0.334565	0.167283	−	0.985909i	$-0.446501\pi$
$937$	9596.00	0.334565	0.167283	+	0.985909i	$0.446501\pi$
$938$	0	0
$939$	0	0
$940$	−1875.00	−0.0650593
$941$	−19782.0	−0.685308	−0.342654	−	0.939462i	$-0.611326\pi$
$941$	−19782.0	−0.685308	−0.342654	+	0.939462i	$0.611326\pi$
$942$	0	0
$943$	−31725.0	−1.09555
$944$	−4260.00	−0.146876
$945$	0	0
$946$	−46440.0	−1.59608
$947$	−4014.00	−0.137738	−0.0688688	−	0.997626i	$-0.521939\pi$
$947$	−4014.00	−0.137738	−0.0688688	+	0.997626i	$0.521939\pi$
$948$	0	0
$949$	−17918.0	−0.612901
$950$	−11175.0	−0.381647
$951$	0	0
$952$	0	0
$953$	−20220.0	−0.687293	−0.343646	−	0.939099i	$-0.611662\pi$
$953$	−20220.0	−0.687293	−0.343646	+	0.939099i	$0.611662\pi$
$954$	0	0
$955$	4590.00	0.155528
$956$	−1638.00	−0.0554150
$957$	0	0
$958$	9234.00	0.311416
$959$	0	0
$960$	0	0
$961$	1893.00	0.0635427
$962$	34503.0	1.15636
$963$	0	0
$964$	1391.00	0.0464742
$965$	4510.00	0.150448
$966$	0	0
$967$	−12244.0	−0.407177	−0.203589	−	0.979057i	$-0.565261\pi$
$967$	−12244.0	−0.407177	−0.203589	+	0.979057i	$0.565261\pi$
$968$	14574.0	0.483911
$969$	0	0
$970$	−20280.0	−0.671290
$971$	57777.0	1.90953	0.954764	−	0.297364i	$-0.0961074\pi$
$971$	57777.0	1.90953	0.954764	+	0.297364i	$0.0961074\pi$
$972$	0	0
$973$	0	0
$974$	−17232.0	−0.566888
$975$	0	0
$976$	−27832.0	−0.912788
$977$	−53910.0	−1.76534	−0.882668	−	0.469997i	$-0.844255\pi$
$977$	−53910.0	−1.76534	−0.882668	+	0.469997i	$0.844255\pi$
$978$	0	0
$979$	10530.0	0.343759
$980$	0	0
$981$	0	0
$982$	6012.00	0.195367
$983$	40059.0	1.29978	0.649890	−	0.760028i	$-0.274815\pi$
$983$	40059.0	1.29978	0.649890	+	0.760028i	$0.274815\pi$
$984$	0	0
$985$	27345.0	0.884552
$986$	−13824.0	−0.446497
$987$	0	0
$988$	−4619.00	−0.148735
$989$	48504.0	1.55949
$990$	0	0
$991$	16670.0	0.534349	0.267175	−	0.963648i	$-0.413910\pi$
$991$	16670.0	0.534349	0.267175	+	0.963648i	$0.413910\pi$
$992$	−8010.00	−0.256369
$993$	0	0
$994$	0	0
$995$	−21680.0	−0.690756
$996$	0	0
$997$	17414.0	0.553166	0.276583	−	0.960990i	$-0.410798\pi$
$997$	17414.0	0.553166	0.276583	+	0.960990i	$0.410798\pi$
$998$	36408.0	1.15478
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.h.1.1		1
3.2	odd	2		735.4.a.g.1.1		1
7.2	even	3		315.4.j.a.46.1		2
7.4	even	3		315.4.j.a.226.1		2
7.6	odd	2		2205.4.a.d.1.1		1
21.2	odd	6		105.4.i.a.46.1	yes	2
21.11	odd	6		105.4.i.a.16.1	✓	2
21.20	even	2		735.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.a.16.1	✓	2	21.11	odd	6
105.4.i.a.46.1	yes	2	21.2	odd	6
315.4.j.a.46.1		2	7.2	even	3
315.4.j.a.226.1		2	7.4	even	3
735.4.a.g.1.1		1	3.2	odd	2
735.4.a.h.1.1		1	21.20	even	2
2205.4.a.d.1.1		1	7.6	odd	2
2205.4.a.h.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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